3.12 \(\int \frac{a+b x}{d-e x^3} \, dx\)

Optimal. Leaf size=161 \[ \frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}} \]

[Out]

-(((b*d^(1/3) - a*e^(1/3))*ArcTan[(d^(1/3) + 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(2/3))) - ((b
*d^(1/3) + a*e^(1/3))*Log[d^(1/3) - e^(1/3)*x])/(3*d^(2/3)*e^(2/3)) + ((b*d^(1/3) + a*e^(1/3))*Log[d^(2/3) + d
^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(2/3))

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Rubi [A]  time = 0.100357, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {1861, 31, 634, 617, 204, 628} \[ \frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}-\frac{\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(d - e*x^3),x]

[Out]

-(((b*d^(1/3) - a*e^(1/3))*ArcTan[(d^(1/3) + 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(2/3)*e^(2/3))) - ((b
*d^(1/3) + a*e^(1/3))*Log[d^(1/3) - e^(1/3)*x])/(3*d^(2/3)*e^(2/3)) + ((b*d^(1/3) + a*e^(1/3))*Log[d^(2/3) + d
^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(2/3)*e^(2/3))

Rule 1861

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 3]], s = Denominato
r[Rt[-(a/b), 3]]}, Dist[(r*(B*r + A*s))/(3*a*s), Int[1/(r - s*x), x], x] - Dist[r/(3*a*s), Int[(r*(B*r - 2*A*s
) - s*(B*r + A*s)*x)/(r^2 + r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && Ne
gQ[a/b]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{a+b x}{d-e x^3} \, dx &=\frac{\left (a+\frac{b \sqrt [3]{d}}{\sqrt [3]{e}}\right ) \int \frac{1}{\sqrt [3]{d}-\sqrt [3]{e} x} \, dx}{3 d^{2/3}}-\frac{\int \frac{\sqrt [3]{d} \left (b \sqrt [3]{d}-2 a \sqrt [3]{e}\right )-\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \sqrt [3]{e} x}{d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 d^{2/3} \sqrt [3]{e}}\\ &=-\frac{\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}-\frac{1}{2} \left (-\frac{a}{\sqrt [3]{d}}+\frac{b}{\sqrt [3]{e}}\right ) \int \frac{1}{d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx+\frac{\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \int \frac{\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{2/3} e^{2/3}}\\ &=-\frac{\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}+\frac{\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}+\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} e^{2/3}}\\ &=-\frac{\left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{2/3}}-\frac{\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} e^{2/3}}+\frac{\left (b \sqrt [3]{d}+a \sqrt [3]{e}\right ) \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0489133, size = 125, normalized size = 0.78 \[ \frac{-\left (a \sqrt [3]{e}+b \sqrt [3]{d}\right ) \left (2 \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )-\log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )\right )-2 \sqrt{3} \left (b \sqrt [3]{d}-a \sqrt [3]{e}\right ) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt{3}}\right )}{6 d^{2/3} e^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(d - e*x^3),x]

[Out]

(-2*Sqrt[3]*(b*d^(1/3) - a*e^(1/3))*ArcTan[(1 + (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] - (b*d^(1/3) + a*e^(1/3))*(2*L
og[d^(1/3) - e^(1/3)*x] - Log[d^(2/3) + d^(1/3)*e^(1/3)*x + e^(2/3)*x^2]))/(6*d^(2/3)*e^(2/3))

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Maple [A]  time = 0.004, size = 188, normalized size = 1.2 \begin{align*} -{\frac{a}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,e}\ln \left ({x}^{2}+\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{b}{6\,e}\ln \left ({x}^{2}+\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-{\frac{b\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(-e*x^3+d),x)

[Out]

-1/3*a/e/(d/e)^(2/3)*ln(x-(d/e)^(1/3))+1/6*a/e/(d/e)^(2/3)*ln(x^2+(d/e)^(1/3)*x+(d/e)^(2/3))+1/3*a/e/(d/e)^(2/
3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x+1))-1/3*b/e/(d/e)^(1/3)*ln(x-(d/e)^(1/3))+1/6*b/e/(d/e)^(1/3)*l
n(x^2+(d/e)^(1/3)*x+(d/e)^(2/3))-1/3*b*3^(1/2)/e/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x+1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-e*x^3+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 6.22848, size = 4563, normalized size = 28.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-e*x^3+d),x, algorithm="fricas")

[Out]

-1/18*(9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sq
rt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))*log(1/36*(9*(I*sqrt
(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*
(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))^2*b*d^2*e - 1/6*(9*(I*sqrt(3) + 1)*
(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b
^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))*a^2*d*e - 2*a*b^2*d - (b^3*d - a^3*e)*x) + 1
/36*(9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + 3*sqrt(1/3)*
sqrt(-((9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*s
qrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))^2*d*e - 144*a*b)/(
d*e)) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))*l
og(-1/36*(9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I
*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))^2*b*d^2*e + 1/6*
(9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3)
+ 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))*a^2*d*e + 2*a*b^2*d - 2*(
b^3*d - a^3*e)*x + 1/12*sqrt(1/3)*((9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/
(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2)
)^(1/3)))*b*d^2*e + 6*a^2*d*e)*sqrt(-((9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*
e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e
^2))^(1/3)))^2*d*e - 144*a*b)/(d*e))) + 1/36*(9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d
- a^3*e)/(d^2*e^2))^(1/3) - 3*sqrt(1/3)*sqrt(-((9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^
3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3
*e)/(d^2*e^2))^(1/3)))^2*d*e - 144*a*b)/(d*e)) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) -
1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))*log(-1/36*(9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(
b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a
^3*e)/(d^2*e^2))^(1/3)))^2*b*d^2*e + 1/6*(9*(I*sqrt(3) + 1)*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a
^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^
2*e^2))^(1/3)))*a^2*d*e + 2*a*b^2*d - 2*(b^3*d - a^3*e)*x - 1/12*sqrt(1/3)*((9*(I*sqrt(3) + 1)*(-1/54*(b^3*d +
a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e)/(
d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))*b*d^2*e + 6*a^2*d*e)*sqrt(-((9*(I*sqrt(3) + 1)*(-1/54*(b^3*
d + a^3*e)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3) + a*b*(-I*sqrt(3) + 1)/(d*e*(-1/54*(b^3*d + a^3*e
)/(d^2*e^2) - 1/54*(b^3*d - a^3*e)/(d^2*e^2))^(1/3)))^2*d*e - 144*a*b)/(d*e)))

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Sympy [A]  time = 1.21121, size = 78, normalized size = 0.48 \begin{align*} - \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{2} - 9 t a b d e - a^{3} e - b^{3} d, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b d^{2} e - 3 t a^{2} d e - 2 a b^{2} d}{a^{3} e - b^{3} d} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-e*x**3+d),x)

[Out]

-RootSum(27*_t**3*d**2*e**2 - 9*_t*a*b*d*e - a**3*e - b**3*d, Lambda(_t, _t*log(x + (9*_t**2*b*d**2*e - 3*_t*a
**2*d*e - 2*a*b**2*d)/(a**3*e - b**3*d))))

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Giac [A]  time = 1.07933, size = 155, normalized size = 0.96 \begin{align*} -\frac{\sqrt{3}{\left (b d^{\frac{2}{3}} e^{\frac{4}{3}} - a d^{\frac{1}{3}} e^{\frac{5}{3}}\right )} \arctan \left (\frac{\sqrt{3}{\left (d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + 2 \, x\right )} e^{\frac{1}{3}}}{3 \, d^{\frac{1}{3}}}\right ) e^{\left (-2\right )}}{3 \, d} - \frac{{\left (b d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + a\right )} e^{\left (-\frac{1}{3}\right )} \log \left ({\left | -d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac{2}{3}}} + \frac{{\left (b d^{\frac{2}{3}} e^{\frac{4}{3}} + a d^{\frac{1}{3}} e^{\frac{5}{3}}\right )} e^{\left (-2\right )} \log \left (d^{\frac{1}{3}} x e^{\left (-\frac{1}{3}\right )} + x^{2} + d^{\frac{2}{3}} e^{\left (-\frac{2}{3}\right )}\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-e*x^3+d),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*(b*d^(2/3)*e^(4/3) - a*d^(1/3)*e^(5/3))*arctan(1/3*sqrt(3)*(d^(1/3)*e^(-1/3) + 2*x)*e^(1/3)/d^(1/
3))*e^(-2)/d - 1/3*(b*d^(1/3)*e^(-1/3) + a)*e^(-1/3)*log(abs(-d^(1/3)*e^(-1/3) + x))/d^(2/3) + 1/6*(b*d^(2/3)*
e^(4/3) + a*d^(1/3)*e^(5/3))*e^(-2)*log(d^(1/3)*x*e^(-1/3) + x^2 + d^(2/3)*e^(-2/3))/d